I came to know that the statement below could be proved using Stallings' binding tie argument, though I have no reference article proving the statement by the binding tie argument. Can anyone help me telling something related to this? Another question, except this, does there exist any other application of binding tie argument, maybe in other dimensions?

Given a $\pi_1$-surjective map $f\colon S\to S'$ between two closed orientable surfaces and a smoothly embedded circle $C$ in $S$, we can homotope $f$ to make it transverse to $C$ so that $f^{-1}(C)$ is either empty or exactly one circle.